The Boro boss had seen his side struggle for points since taking charge but they had this match virtually sewn up before the interval.
Injury-hit Rovers were their own worst enemies as they made uncharacteristic mistakes in defence.
But Boro deserved credit for their tremendous work-rate from the first whistle as they knocked the normally slick passing home side out of their stride by closing them down quickly and refusing to allow them to settle on the ball.
When they did have possession Doncaster looked dangerous and Mark Wilson was denied by Danny Coyne from close range in the fifth minute.
Both teams tended to cancel each other out in a first half played at 100 miles per hour but Doncaster threatened again when Lewis Guy had a shot blocked in the 23rd minute.
Billy Sharp saw another Doncaster effort deflected just wide a couple of minutes later before Middlesbrough struck with a goal completely out of the blue.
There seemed to be no danger when highly-rated winger Adam Johnson latched onto the ball 25 yards out but he unexpectedly lashed home a cracking shot to put Boro in front.
Strachan's men stunned Rovers again when Johnson pulled the ball back for Jonathan Franks to fire home on the stroke of half-time.
Worse was to follow for Doncaster when a bad mix-up between keeper Neil Sullivan and defender James O'Connor led to Leroy Lita adding a third in the 66th minute.
Teenage midfielder Jordon Mutch marked his debut for Rovers after signing on-loan from Birmingham City with a magnificent 30-yard strike in the 83rd minute to give the home side some consolation.
But it was Boro that had the last word when the impressive Johnson drilled home a 20-yard free-kick in stoppage time.